Differential-difference operators and monodromy representations of Hecke algebras
type: Item
instance of: scholarly article
title: Differential-difference operators and monodromy representations of Hecke algebras
zbMATH Open document ID: 0821.33009
author: Charles F. Dunkl
published in: Pacific Journal of Mathematics
publication date: 1994-03-01T00:00:00Z
review text: Associated to any finite reflection group \(G\) on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections. The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-difference operator in the algebra. For each irreducible representation of \(G\) the differential-difference equations lead to a linear system of first-order meromorphic differential equations corresponding to an integrable connection over the \(G\)-orbits of regular points in the complexification of the Euclidean space. The fundamental group is the generalized Artin braid group belonging to \(G\), and its monodromy representation factors over the Hecke algebra of \(G\). Monodromy has long been of importance in the study of special functions of several variables, for example, the hyperlogarithms of Lappo-Danilevsky are used to express the flat sections and the work of Riemann on the monodromy of the hypergeometric equation is applied to the case of dihedral groups. The case of dihedral groups with two conjugacy classes was subsequently solved by the author in [Integral Transforms Spec. Funct. 1, 75-86 (1993; Zbl 0821.33008)].
reviewed by: Charles F. Dunkl
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 509118
zbMATH Keywords: reflection group
zbMATH Keywords: differential-difference operators
zbMATH Keywords: Hecke algebra
zbMATH Keywords: monodromy action
zbMATH Keywords: Artin braid group
MaRDI profile type: Publication
full work available at URL: pjm.1993.159.271
OpenAlex ID: W1996195710
DOI: 10.2140/PJM.1993.159.271
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description: scientific article; zbMATH DE number 509118